1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

BASIC RICCI FLOW (3) The Ricci tensor Re of g evolves by (Vl-6.7) :t Rjk = b..LRjk = b..Rjk + 2gPqgrs RpjkrRqs - 2gpq RjpRqk· ...

458 A. BASIC RICCI FLOW THEORY THEOREM A.17 (Short-time existence on noncompact manifolds). Let M be a noncompact manifold and l ...

BASIC RICCI FLOW PROOF. Let p (t) ~ Rmin (t). Under the unnormalized Ricci flow, dp > 3._p2 > 0. dt - n - Under the norm ...

460 A. BASIC RICCI FLOW THEORY LEMMA A.23. If we evolve the isometry l (t) by (Vl-6.19a) (Vl-6.19b) then the bundle maps remain ...

BASIC RICCI FLOW 461 (See pp. 187-189 in Section 4 of Chapter 6 in Volume One.) The PDE (Vl-6.27) governing the behavior of Rm ...

462 A. BASIC RICCI FLOW THEORY THEOREM A.27 (RF on closed 3-orbifolds with Re> 0). If (V^3 ,go) is a closed Riemannian 3-orbi ...

BASIC RlCCI FLOW 463 So if go has positive Ricci curvature, then there exist constants C < oo and 8 > 0 such that (A.23) ...

464 A. BASIC RICCI FLOW THEORY then (A.24) I ( )I C (n, K, r, a) _ C(n, JKr, a)K V'Rm y,t :::; Vt - Vt for all (y,t) E Bg(O) (p, ...

3. Basic singularity theory for Ricci fl.ow Basic singularity theory for Ricci fl.ow The knowledge of which geometry aims is t ...

466 A. BASIC RICCI FLOW THEORY • one says (M, g (t)) forms a Type IV singularity as t-> 0 if sup t !Rm(·, t)I < oo. Mx(O,T ...

BASIC SINGULARITY THEORY FOR RICCI FLOW 467 3.3. Trace Harnack inequality. Given a surface (M^2 , g) with posi- tive curvature ...

468 A. BASIC RICCI FLOW THEORY 3.4. Surface entropy formulas. The surface entropy N is defined for a metric of strictly positive ...

BASIC SINGULARITY THEORY FOR RICCI FLOW 469 solution or sausage model (see [311] or [141]) of the Ricci fl.ow is the metric g ...

470 A. BASIC RICCI FLOW THEORY 3.6. Necklike points in Type I solutions. (See Section 4 in Chap- ter 9 of Volume One.) We say th ...

MORE RICCI FLOW THEORY AND ANCIENT SOLUTIONS 471 THEOREM A.53 (Strong maximum principle for Rm). Let (Mn,g(t)), t E [O, T), be ...

472 A. BASIC RICCI FLOW THEORY The following is a generalization of the trace Harnack estimate (see [105] and [290]). THEOREM A. ...

MORE RICCI FLOW THEORY AND ANCIENT SOLUTIONS 473 Theorem A.58 says the following. (See Chapter 6 for a definition of r;;-nonco ...

474 A. BASIC RICCI FLOW THEORY PROPOSITION A.66 (n 2:: 2 backward limit of Type II ancient solution with Rm 2:: 0 and sect> 0 ...

CLASSICAL SINGULARITY THEORY 475 THEOREM A.69 (3d Type I - existence of necks). If (M^3 ,g (t)) is a Type I singular solution ...

476 A. BASIC RICCI FLOW THEORY equal to infinity. Thus we can apply dimension reduction, Theorem A.59, to get a second limit whi ...